Good question! I believe the one in Bishop is the mothership. It has a little more going on (selection, crowds, size, etc), but I think they are basically the same (i.e. same owners, stuff is trucked from one location to the other, etc).

Well, whenever I go into the Bishop one, right at the register, there's a big, unmistakable sign saying they are in no way associated with the Mammoth one (sort of like an East Of Eden family feud).

One factor in favor of doing Split Mt. is that it's infinitely easier, yet probably more scenic to climb that from the JMT than it is from the east. So if you think you'll want to check that somewhat isolated 14er off your list someday, making the short climb from the JMT is the way to go. (As opposed to something like Tyndall, which you may want to hit on the same trip as climbing Williamson someday.)